Proportions: shares
A proportion is the statistic best suited to test research hypotheses addressing the share of a category or entity in the population. The hypothesis that a television station reaches half of all households in a country provides an example. All households in the country constitute the population. The share of the television station is the proportion or percentage of all households watching this television station.
If we want to use a statistic, we need to know the variable and cases (units of analysis) for which the statistic must be calculated. In this example, a household does or does not watch the television station, so our variable is a dichotomy with the two categories (“No, does not watch this station”, “Yes, watches this station”) usually coded as 0 versus 1 or 1 versus 2.
Each household provides an observation, namely either the score 0 or the score 1 on this variable or no score if there are missing values. To test the research hypothesis that a television station reaches half of all households in a country, we have to formulate a statistical hypothesis about the proportion—of households viewing this television station—in the population—all households in this country. For example, the researcher’s statistical hypothesis could be that the proportion in the population is 0.5.
We can also be interested in more than two categories, for instance, does the television station reach the same share of all households in the north, east, south, and west of the country? This translates into a statistical hypothesis containing three or more proportions in the population. If 30% of households in the population are situated in the west, 25 % in the south and east, and 20% in the north, we would expect these proportions in the sample if all regions are equally represented. Our statistical hypothesis is actually a relative frequency distribution, such as, for instance, in Table 9.3.
| Region | Hypothesized Proportion |
|---|---|
| North | 0.20 |
| East | 0.25 |
| South | 0.25 |
| West | 0.30 |
A test for this type of statistical hypothesis is called a one-sample chi-squared test. It is up to the researcher to specify the hypothesized proportions for all categories. This is not a simple task: What reasons do we have to expect particular values, say a region’s share of thirty per cent of all households instead of twenty-five per cent?
The test is mainly used if researchers know the true proportions of the categories in the population from which they aimed to draw their sample. If we try to draw a sample from all citizens of a country, we usually know the frequency distribution of sex, age, educational level, and so on for all citizens from the national bureau of statistics. With the bureau’s information, we can test if the respondents in our sample have the same distribution with respect to sex, age, or educational level as the population from which we tried to draw the sample; just use the official population proportions in the null hypothesis.
If the proportions in the sample do not differ more from the known proportions in the population than we expect based on chance, the sample is representative of the population in the statistical sense (see Section 1.2.6). As always, we use the p value of the test as the probability of obtaining our sample or a sample that is even more different from the null hypothesis, if the null hypothesis is true. Note that the null hypothesis now represents the (distribution in) the population from which we tried to draw our sample. We conclude that the sample is representative of this population in the statistical sense if we can not reject the null hypothesis, that is, if the p value is larger than .05. Not rejecting the null hypothesis means that we have sufficient probability that our sample was drawn from the population that we wanted to investigate. We can now be more confident that our sample results generalize to the population that we meant to investigate.
Testing proportions in SPSS
Figure 9.18: A binomial test on a single proportion in SPSS.
Figure 9.19: A one-sample t test in SPSS.
For a binomial test, see the video in Figure 9.18. A one-sample chi-squared test is explained in the video of Figure 9.20.
Figure 9.20: A chi-squared test on a frequency distribution in SPSS.